My uninformed opinion is that games should be a good medium for teaching young children. (I would be very interested in hearing opinions about this from people who actually work in education.) I think a surprising amount can be learned by carefully inspecting even a game as simple as Minesweeper. There are various "lemmas" you can prove in Minesweeper about how various configurations uniquely pin down the location of mines or non-mines and I suspect this could be an accessible introduction to mathematical proof (the advantage being that all of the relevant proofs can be done by analyzing a finite number of cases). I haven't actually tried this though.
I agree and would emphasize that deriving concepts from an existing game is preferable to constructing an educational game from scratch. It makes it more engaging and teaches the skill of modelling.
What games do children that age play nowadays?
In about eight months or so, I will be one of those (hopefully not starving) students. I'll be moving out to London to live with my aunt and uncle in a rather nice middle-class neighbourhood, while I study and work to prepare for university the following year. They know a lot of the parents around there and suggested that I begin teaching small groups of 8-to-12 year old children for maybe an hour or two regularly, and charge their parents/guardians a reasonable sum per child. I would be teaching them math and science in all likelihood. Apparently word will get around quickly if I'm competent so I might have a substantial number of customers within a few months.
My questions: